Integration by substitution

1. Compute each indefinite integral.

(a) \( \int (1-2x)^5  dx\)  solution

(b) \(\displaystyle{\int \frac{x^2}{1+x^3}\, dx}\)  solution

(c) \(\displaystyle{\int x^2 (x^3-2)^7 \, dx}\)  solution

(d) \( \displaystyle{\int \frac{x}{\frac{1}{2}x^2-5}}\, dx \)  solution

(e) \( \int \displaystyle{ e^{-x} \, dx }\)  solution

(f) \( \int\displaystyle{e^{3x-1}\, dx}\)   solution

(g) \( \displaystyle{\int x \,\sqrt{1-x^2}   \, \,  dx}\)  Solution

(h)\( \displaystyle{\int (t-2)\sqrt{3t^2-12t} \, \,  dt}\)   Solution

2. Find the indefinite integral.

(i) \(\displaystyle{\int e^{x^5} x^4\, dx}\)  solution

(ii) \(\displaystyle{\int \frac{x^2-4}{x^3-12x+6} \, dx}\)  solution

(iii) \(\displaystyle{\int \frac{x^8}{\sqrt{x^9-7}}\, dx}\)   solution

(iv) \(\displaystyle{\int \frac{e^{5x}}{2+e^{5x}} \, dx}\)  solution

3. Compute the following definite integral.

  1. \(\displaystyle{ \int_{-1}^1  x\,(x^2+1)^2  \, \,  dx}\)  Solution 

  2. \(\displaystyle{ \int_0^1  x \, \sqrt{5x^2+4}  \, \, dx}\)  Solution 

  3. \(\displaystyle{ \int_1^2  \frac{x}{(x^2+3)^2}  \, \,   dx}\)   Solution 

  4. \( \displaystyle{\int_0^1 \frac{x+2}{\sqrt{x^2+4x}}  \, \,    dx}\)    Solution

  5. \( \displaystyle{\int_0^4 \frac{e^{\sqrt{x}}} {\sqrt{x}}   \, \,   dx} \)   Solution